Problem: Add the following rational expressions. $\dfrac{z^2}{z^3+4}+\dfrac{4z^2-6z}{z^3+4}=$
We want to add two rational expressions whose denominators are equal. We can do this by adding the numerators and keeping the denominator the same. [Does this fit with how we add rational numbers?] $\begin{aligned} &\phantom{=}\dfrac{z^2}{z^3+4}+\dfrac{4z^2-6z}{z^3+4} \\\\ &=\dfrac{(z^2)+(4z^2-6z)}{z^3+4} \\\\ &=\dfrac{z^2+4z^2-6z}{z^3+4} \\\\ &=\dfrac{5z^2-6z}{z^3+4} \end{aligned}$ In conclusion, $\dfrac{z^2}{z^3+4}+\dfrac{4z^2-6z}{z^3+4}=\dfrac{5z^2-6z}{z^3+4}$